Lie symmetry analysis, particular solutions and conservation laws for the dissipative (2 + 1)- dimensional AKNS equation

Sixing Tao; Sixing Tao

doi:10.3934/cam.2023024

Communications in Analysis and Mechanics

2023, Volume 15, Issue 3: 494-514. doi: 10.3934/cam.2023024

Previous Article Next Article

Research article Special Issues

Lie symmetry analysis, particular solutions and conservation laws for the dissipative (2 + 1)- dimensional AKNS equation

Sixing Tao ^,

School of Mathematics and Statistics, Shangqiu Normal University, Shangqiu 476000, China

Received: 01 June 2023 Revised: 20 July 2023 Accepted: 09 August 2023 Published: 17 August 2023
35A25, 35G50, 35Q35, 37K10

The dissipative (2 + 1)-dimensional AKNS equation is considered in this paper. First, the Lie symmetry analysis method is applied to the dissipative (2 + 1)-dimensional AKNS and six point symmetries are obtained. Symmetry reductions are performed by utilizing these obtained point symmetries and four differential equations are derived, including a fourth-order ordinary differential equation and three partial differential equations. Thereafter, the direct integration approach and the $ (G'/G^{2})- $expansion method are employed to solve the ordinary differential respectively. As a result, a periodic solution in terms of the Weierstrass elliptic function is obtained via the the direct integration approach, while six kinds of including the hyperbolic function types and the hyperbolic function types are derived via the $ (G'/G^{2})- $expansion method. The corresponding graphical representation of the obtained solutions are presented by choosing suitable parametric values. Finally, the multiplier technique and the classical Noether's theorem are employed to derive conserved vectors for the dissipative (2 + 1)-dimensional AKNS respectively. Consequently, eight local conservation laws for the dissipative (2 + 1)-dimensional AKNS equation are presented by utilizing the multiplier technique and five local conservation laws are derived by invoking Noether's theorem.
- Lie symmetry analysis,
- the dissipative (2 + 1)-dimensional AKNS equation,
- conservation laws,
- the multiplier technique,
- Noether's theorem
Citation: Sixing Tao. Lie symmetry analysis, particular solutions and conservation laws for the dissipative (2 + 1)- dimensional AKNS equation[J]. Communications in Analysis and Mechanics, 2023, 15(3): 494-514. doi: 10.3934/cam.2023024

Related Papers:

Abstract

The dissipative (2 + 1)-dimensional AKNS equation is considered in this paper. First, the Lie symmetry analysis method is applied to the dissipative (2 + 1)-dimensional AKNS and six point symmetries are obtained. Symmetry reductions are performed by utilizing these obtained point symmetries and four differential equations are derived, including a fourth-order ordinary differential equation and three partial differential equations. Thereafter, the direct integration approach and the $ (G'/G^{2})- $expansion method are employed to solve the ordinary differential respectively. As a result, a periodic solution in terms of the Weierstrass elliptic function is obtained via the the direct integration approach, while six kinds of including the hyperbolic function types and the hyperbolic function types are derived via the $ (G'/G^{2})- $expansion method. The corresponding graphical representation of the obtained solutions are presented by choosing suitable parametric values. Finally, the multiplier technique and the classical Noether's theorem are employed to derive conserved vectors for the dissipative (2 + 1)-dimensional AKNS respectively. Consequently, eight local conservation laws for the dissipative (2 + 1)-dimensional AKNS equation are presented by utilizing the multiplier technique and five local conservation laws are derived by invoking Noether's theorem.

References

[1]	C. M. Khalique, A. Biswas, A Lie symmetry approach to nonlinear Schrödinger's equation with non-Kerr law nonlinearity, Commun. Nonlinear Sci. Numer. Simulat., 14 (2009), 4033–4040. https://doi.org/10.1016/j.cnsns.2009.02.024 doi: 10.1016/j.cnsns.2009.02.024
[2]	J. J. Mao, S. F. Tian, T. T. Zhang, X. J. Yan, Lie symmetry analysis, conservation laws and analytical solutions for chiral nonlinear Schrödinger equation in (2 +1)-dimensions, Nonlinear Anal. Model Control, 25(2020), 358–377. https://doi.org/10.15388/namc.2020.25.16653 doi: 10.15388/namc.2020.25.16653
[3]	N. Benoudina, Y. Zhang, C. M. Khalique, Lie symmetry analysis, optimal system, new solitary wave solutions and conservation laws of the Pavlov equation, Commun. Nonlinear Sci. Numer. Simulat., 94 (2021), 105560. https://doi.org/10.1016/j.cnsns.2020.105560 doi: 10.1016/j.cnsns.2020.105560
[4]	C. H. Gu, H. S. Hu, A unified explicit form of Bäcklund transformations for generalized hierarchies of the KdV equation, Lett. Math. Phys., 11 (1986), 325–337. https://doi.org/10.1007/BF00574158 doi: 10.1007/BF00574158
[5]	W. X. Ma, Y. J. Zhang, Darboux transformations of integrable couplings and applications, Rev. Math. Phys., 30 (2018), 1850003. https://doi.org/10.1142/S0129055X18500034 doi: 10.1142/S0129055X18500034
[6]	G. Q. Xu, A. M. Wazwaz, Bidirectional solitons and interaction solutions for a new integrable fifth-order nonlinear equation with temporal and spatial dispersion, Nonlinear Dyn., 101 (2020), 581–595. https://doi.org/10.1007/s11071-020-05740-1 doi: 10.1007/s11071-020-05740-1
[7]	M. L. Wang, X. Z. Li, J. L. Zhang, The $(G'/G)$-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A, 372 (2008), 417–423. https://doi.org/10.1016/j.physleta.2007.07.051 doi: 10.1016/j.physleta.2007.07.051
[8]	S. Sirisubtawee, S. Koonprasert, Exact traveling wave solutions of certain nonlinear partial differential equations using the $(G'/G^{2})$-expansion method, Adv. Math. Phys., 2018 (2018), 7628651. https://doi.org/10.1155/2018/7628651 doi: 10.1155/2018/7628651
[9]	S. X. Tao, Breathers, resonant multiple waves and complexiton solutions of a (2+1)-dimensional nonlinear evolution equation, AIMS Math., 8 (2023), 11651–11665. https://doi.org/10.3934/math.2023590 doi: 10.3934/math.2023590
[10]	Y. D. Zhuang, Y. Zhang, H. Y. Zhang, P. Xia, Multi-soliton solutions for the three types of nonlocal hirota equations via riemann–hilbert approach, Commun. Theor. Phys., 74 (2022), 115004. https://doi.org/10.1088/1572-9494/ac8afc doi: 10.1088/1572-9494/ac8afc
[11]	C. M. Khalique, L. D. Moleleki, A (3+ 1)-dimensional generalized BKP-Boussinesq equation: Lie group approach, Results Phys., 13 (2019), 102239. https://doi.org/10.1016/j.rinp.2019.102239 doi: 10.1016/j.rinp.2019.102239
[12]	I. Simbanefayi, C. M. Khalique, Travelling wave solutions and conservation laws for the Korteweg-de Vries-Bejamin-Bona-Mahony equation, Results Phys., 8 (2018), 57–63. https://doi.org/10.1016/j.rinp.2017.10.041 doi: 10.1016/j.rinp.2017.10.041
[13]	H. Z. Liu, L. J. Zhang, Symmetry reductions and exact solutions to the systems of nonlinear partial differential equations, Phys. Scr., 94 (2019), 015202. https://doi.org/10.1088/1402-4896/aaeeff doi: 10.1088/1402-4896/aaeeff
[14]	N. Benoudina, Y. Zhang, N. Bessaad, A new derivation of (2 + 1)-dimensional Schrödinger equation with separated real and imaginary parts of the dependent variable and its solitary wave solutions, Nonlinear Dyn., 111 (2023), 6711–6726. https://doi.org/10.1007/s11071-022-08193-w doi: 10.1007/s11071-022-08193-w
[15]	S. X. Tao, Lie symmetry analysis, particular solutions and conservation laws of a (2+1)- dimensional KdV4 equation, Math. Bioci. Eng., 20 (2023), 11978–11997. https://doi.org/10.3934/mbe.2023532 doi: 10.3934/mbe.2023532
[16]	N. Benoudina, Y. Zhang, C. M. Khalique, N Bessaad, Novel hybrid solitary waves and shrunken-periodic solutions, solitary Moiré pattern and conserved vectors of the (4+1)-Fokas equation, Int. J. Geom. Methods Mod. Phys., 19 (2022), 2250195. https://doi.org/10.1142/S021988782250195X doi: 10.1142/S021988782250195X
[17]	A. H. Bokhari, A. Y. Al-Dweik, A. H. Kara, F. M. Mahomed, F. D. Zaman, Double reduction of a nonlinear (2 + 1) wave equation via conservation laws, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 1244–1253. https://doi.org/10.1016/j.cnsns.2010.07.007 doi: 10.1016/j.cnsns.2010.07.007
[18]	G. L. Caraffini, M. Galvani, Symmetries and exact solutions via conservation laws for some partial differential equations of mathematical physics, Appl. Math. Comput., 219 (2012), 1474–1484. https://doi.org/10.1016/j.amc.2012.07.050 doi: 10.1016/j.amc.2012.07.050
[19]	E. Noether, Invariant variation problems, Transp. Theory Stat. Phys., 1 (1971), 186–207. https://doi.org/10.1080/00411457108231446 doi: 10.1080/00411457108231446
[20]	W. Sarlet, Comment on 'Conservation laws of higher order nonlinear PDEs and the variational conservation laws in the class with mixed derivatives', J. Phys. A: Math. Theor., 43 (2010), 458001. https://doi.org/10.1088/1751-8113/43/45/458001 doi: 10.1088/1751-8113/43/45/458001
[21]	P. J. Olver, Applications of Lie Groups to Differential Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1993. https://doi.org/10.1007/978-1-4684-0274-2
[22]	N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 333 (2007), 311–328. https://doi.org/10.1016/j.jmaa.2006.10.078
[23]	C. M. Khalique, S. A. Abdallah, Coupled Burgers equations governing polydispersive sedimentation; a Lie symmetry approach, Results Phys., 16 (2020), 102967. https://doi.org/10.1016/j.rinp.2020.102967 doi: 10.1016/j.rinp.2020.102967
[24]	A. M. Wazwaz, N-soliton solutions for shallow water waves equations in (1+1) and (2+1) dimensions, Appl. Math. Comput., 217 (2011), 8840–8845. https://doi.org/10.1016/j.amc.2011.03.048 doi: 10.1016/j.amc.2011.03.048
[25]	M. Najafi, M. Najafi, M. T. Darvishi, New Exact Solutions to the (2+1)-Dimensional Ablowitz-Kaup-Newell-Segur Equation: Modification of the Extended Homoclinic Test Approach, Chin. Phys. Lett., 29 (2012), 040202. https://doi.org/10.1088/0256-307X/29/4/040202 doi: 10.1088/0256-307X/29/4/040202
[26]	Q. Liu, W. G. Zhang, Exact travelling wave solutions for the dissipative (2+1)-dimensional AKNS equation, Appl. Math. Comput., 217 (2010), 735–744. https://doi.org/10.1016/j.amc.2010.06.011 doi: 10.1016/j.amc.2010.06.011
[27]	N. Liu, X. Q. Liu, Application of the binary Bell polynomials method to the dissipative (2+1)-dimensional AKNS equation, Chin. Phys. Lett., 29 (2012), 120201. https://doi.org/10.1088/0256-307X/29/12/120201 doi: 10.1088/0256-307X/29/12/120201
[28]	Z. L. Cheng, X. H. Hao, The periodic wave solutions for a (2 +1)-dimensional AKNS equation, Appl. Math. Comput., 234 (2014), 118–126. https://doi.org/10.1016/j.amc.2014.01.082 doi: 10.1016/j.amc.2014.01.082
[29]	H. Wang, Y. H. Wang, CRE solvability and soliton-cnoidal wave interaction solutions of the dissipative (2+ 1)-dimensional AKNS equation, Appl. Math. Lett., 69 (2017), 161–167. https://doi.org/10.1016/j.aml.2017.02.007 doi: 10.1016/j.aml.2017.02.007
[30]	W. X. Ma, Application of the Riemann–Hilbert approach to the multicomponent AKNS integrable hierarchies, Nonlinear Anal. Real World Appl., 47 (2019), 1–17. https://doi.org/10.1016/j.nonrwa.2018.09.017 doi: 10.1016/j.nonrwa.2018.09.017
[31]	H. C. Ma, Y. D. Gao, A. P. Deng, Dynamical analysis of diversity lump solutions to the (2+1)-dimensional dissipative Ablowitz-Kaup-Newell-Segure equation, Commun. Theor. Phys., 74 (2022), 115003. https://doi.org/10.1088/1572-9494/ac633f doi: 10.1088/1572-9494/ac633f
[32]	Z. Y. Ma, H. L. Wu, Q. Y. Zhu, Lie symmetry, full symmetry group and exact solution to the (2+ 1)-dimemsional dissipative AKNS equation, Rom. J. Phys., 62 (2017), 114.

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)